The common cold is an infection caused by a number of more than 200 virusses.
The rhinoviruses are the most common culprits implicated in the cold.
Other viruses include human coronavirus, influenza viruses, adenoviruses, parainfluenza viruses, human respiratory syncytial virus, enteroviruses other than rhinoviruses, and metapneumovirus.
Often more than one virus is present.
The symptoms e.g. runny nose etc are not caused by the virus itself however but indeed by the body's own defense systems, much like a response to an irritation.
Next: Differences between the Common Cold and the Flu (Influenza)
References:
HealthlineWikipedia